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We elaborate on statistical thermodynamics models of relativistic geometricflows as generalisations of G. Perelman and R. Hamilton theory centred aroundC. Carath\'{e}odory axiomatic approach to thermodynamics with Pfaffiandifferential equations. The anholonomic frame deformation method, AFDM, forconstructing generic off--diagonal and locally anisotropic cosmologicalsolitonic solutions in the theory of relativistic geometric flows and generalrelativity is developed. We conclude that such solutions can not be describedin terms of the Hawking--Bekenstein thermodynamics for hypersurface,holographic, (anti) de Sitter and similar configurations. The geometricthermodynamic values are defined and computed for nonholonomic Ricci flows,(modified) Einstein equations, and new classes of locally anisotropiccosmological solutions encoding solitonic hierarchies.

Constantin Carath\'eodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions